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MNRAS 000, 1–9 (2020) Preprint 4 September 2020 Compiled using MNRAS LATEX style file v3.0

The influence of line opacity treatment in STELLA on

supernova light curves

Alexandra Kozyreva 1,2⋆, Luke Shingles 3, Alexey Mironov 4

Petr Baklanov 5,6,7 Sergey Blinnikov 4,5,6,81Max-Planck-Institut fur Astrophysik, Garching bei Munchen, 85748, Germany,2Alexander von Humboldt Fellowship3Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, UK4Sternberg astronomical institute of Lomonosov Moscow State University, 119992, Moscow, Russia5Space Research Institute (IKI), 84/32 Profsoyuznaya, Moscow, 117997, Russia6NRC “Kurchatov Institute” – ITEP, Moscow, 117218, Russia7National Research Nuclear University Moscow Engineering Physics Institute, Moscow, 115409, Russia8 Dukhov Automatics Research Institute (VNIIA), 127055 Moscow, 127055, Russia

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT

We systematically explore the effect of the treatment of line opacity on supernova lightcurves. We find that it is important to consider line opacity for both scattering andabsorption (i.e. thermalisation which mimics the effect of fluorescence.) We explorethe impact of degree of thermalisation on three major types of supernovae: Type Ia,Type II-peculiar, and Type II-plateau. For that we use radiative transfer code STELLAand analyse broad-band light curves in the context of simulations done with the spec-tral synthesis code ARTIS and in the context a few examples of observed supernovaeof each type. We found that the plausible range for the ratio between absorption andscattering in the radiation hydrodynamics code STELLA is (0.8-1):(0.2–0), i.e. therecommended thermalisation parameter is 0.9.

Key words: supernovae: general – supernovae – stars: massive – radiative transfer

1 MOTIVATION

Large sets of observational data have become available fromsupernova-search surveys and transient robotic systems.Among these data, there are dozens of discovered super-novae (SNe) with spectral snapshots spanning the earliestepochs up to hundreds of days after explosion. The compar-ison between observed spectral evolution of SNe and numer-ical simulations provides clues about the progenitor systemsand the explosion mechanisms. One of the tasks for theoret-ical studies using radiative transfer simulations is to repro-duce the evolution of the radiation field in the fast movingSN ejecta and the energy distribution across a wide spectralrange. A number of sophisticated radiative transfer codes inthe literature are used to carry out detailed spectral synthe-sis simulations (Mazzali & Lucy 1993; Baron et al. 1996a;Kasen et al. 2006; Dessart & Hillier 2010; Jerkstrand et al.2011; Wollaeger et al. 2013, and many others). Othercodes do not simulate spectra but assume approxiatetreatments for the formation of lines and the redistribu-

⋆ E-mail: sasha@mpa-garching.mpg.de

tion of energy resulting from radiation–matter interaction(Blinnikov et al. 1998; Bersten et al. 2011; Piro & Morozova2014; Utrobin et al. 2015, and others).

In the current study, we discuss a particular aspect ofradiative transfer simulations for SNe that reflects the micro-physics of photon–atom interaction, namely the probabilityfor photons to be either resonantly scattered or inelasticallyabsorbed. For the sophisticated codes from the first group,there is no simplification assumed. In the second group ofcodes, these processes are treated approximately with a ther-malisation parameter that determines the ratio of scatteringto absorption. In this paper, we discuss the best value to usefor the thermalisation parameter in the hydrodynamics ra-diative transfer code STELLA by comparison to observed SNeand to advanced spectral synthesis codes.

In Table 1, we list several radiative transfer codes andtheir assumed values for the thermalisation parameter. Notethat a few papers using SEDONA state different values of thethermalisation parameter, while the standard default valueis ε = 0.9 for recent studies (Nathaniel Roth, Daniel Kasen,private communication). The choice of pure absorption lineopacity can be justified on the basis that the strongest lines

© 2020 The Authors

2 A. Kozyreva et al.

Table 1. The choice of thermalisation parameter in different radiative transfer codes. Note, that Baron et al. (1996b) apply ε to a seriesof lines considered in LTE while doing full non-LTE radiative transfer. Goldstein & Kasen (2018) apply ε = 0 for elements with atomicnumber Z ≤ 20, and ε = 1 for Z > 20. See text below for details about the choice of the thermalisation parameter in Blinnikov et al.(1998).

Reference Absorption Scattering Code name Application Details

Baron et al. (1996b) 0.05–0.1 0.9-0.95 PHOENIX SNe Ia, II Non-LTE with a number of LTE linesNugent et al. (1997) 0.1 0.9 PHOENIX SNe Ia as aboveBlinnikov et al. (1998) 0/1 1/0 STELLA SNe I, II LTE; no temperature for radiationKromer & Sim (2009) – – ARTIS SNe I, IIDessart & Hillier (2010) – – CMFGEN SNe I, II full Non-LTE from kinetic equationsKasen et al. (2006) 0.3-1 0.7-0 SEDONA SNe Ia, II LTEGoldstein & Kasen (2018) 1/0 0/1 SEDONA SNe Ia as aboveShen et al. (2018) 1 0 SEDONA SNe Ia as aboveUtrobin et al. (2015) 0 1 CRAB SNe II LTE with corrections for Non-LTE, grey atmosphereMagee et al. (2018) 0.9 0.1 TURTLS early SNe Ia LTE

are iron lines, and detailed studies (e.g. Kasen 2006) showthat the iron lines are mostly purely absorptive. Althoughphotons are not immediately thermalised, the true thermali-sation timescale is extremely short (Pinto & Eastman 2000).Initial high-energy photons are absorbed and re-emittedat longer wavelengths, i.e. the effect of fluorescence occursbroadly in the SN ejecta (e.g. Hoflich 1995; Lucy 1999;Pinto & Eastman 2000). Goldstein & Kasen (2018) suggesttreating all lines from elements with atomic numbers be-low 20 (Z ≤ 20) as “purely scattering” and all lines fromelements with atomic numbers above 20 as “purely absorp-tive”. Kasen (2006) analysed the effect of Ca II triplet on I

band light curve and concluded that Ca must be treated aspurely scattering, otherwise the I magnitude does not matchobserved light curves, in particular at the second maximum.

In the present study, we address the accuracy of lineopacity treatment in the hydrodynamics radiative transfercode STELLA. In the basic descriptive paper about STELLA,the authors raised the issue of the choice between scat-tering and absorptive treatment of lines (Blinnikov et al.1998). The STELLA light curves were compared to those cal-culated with EDDINGTON (Eastman & Pinto 1993), which is afull Non-LTE (no assumed Local Thermodynamic Equilib-rium) radiative transfer code. Note that simulations with theLTE option and forced absorptive line opacity in EDDINGTON

were used for that comparison analysis. Blinnikov et al.(1998) concluded that the lines in STELLA ought to beabsorption-dominated in order to provide better agreementwith EDDINGTON and with the observed SN1993J. Hence,the standard value of thermalisation parameter in STELLA

is ε = 1 since 1998. More recently, STELLA is now the part ofthe latest MESA

1 release (Paxton et al. 2018) which is pub-licly available. STELLA operates under the assumption thatthermalisation parameter is applied to all species, for alltransitions, and independent of electron density, which isindeed a major simplification. However, the advantage ofSTELLA is that it is a hydrodynamics code, i.e., it solvesimplicitly coupled hydrodynamics and multigroup radiationtransport. This enables STELLA to accurately capture shockpropagation if the option for artificial explosion (thermal orkinetic bomb) is set in MESA. Among other advantages is the

1 Modules for Experiments in Stellar Astrophysicshttp://mesa.sourceforge.net/ (Paxton et al. 2011, 2013,2015, 2018).

ability of STELLA to provide reliable predictions for photo-metric properties of SN explosions without requiring largecomputational resources.

The paper is organised as follows: We describe themethod and models in Section 2. In Section 3, we discussour procedure of calibration of the thermalisation parame-ter in STELLA using representative models of SN Ia, SN IIpec,and SN IIP. We finalise the analysis in Section 4, where wespecify the recommended value for the thermalisation pa-rameter in STELLA.

2 INPUT MODELS AND METHOD

For the current study, we used three models from the liter-ature. The goal is to explore the impact of different degreesof thermalisation of lines for application to:

(i) normal SNe Ia — the basic W7 model (Nomoto et al.1984).

(ii) SNe IIpec — the 16-7b model from Menon & Heger(2017); Menon et al. (2019) exploded with the explosion en-ergy of 2.33 foe (final kinetic energy 1.9 foe).

(iii) normal SNe IIP — the L15 model from Limongi et al.(2000); Utrobin et al. (2017) exploded with the explosionenergy of 1.1 foe (final kinetic energy 0.74 foe);

These three models are not universal for the three types ofSNe we discuss, but serve as reference models for each type.

One of the main diagnostics for a possible ratio be-tween scattering and absorptive line opacity is the inspectionof spectra, specifically how lines redistribute energy withinspectral bands. STELLA solves radiative transfer equations ina standard 100 frequency bins, which are not enough to con-struct detailed spectra. However, the code provides spectralenergy distributions (hereafter, SED) which allow us to inte-grate flux in standard BESSEL broad bands. Therefore, therestriction on the thermalisation parameter might be deter-mined through analysis of the broad band magnitudes. Be-low we analyse separately three SN types: SNe Ia, SNe IIpec,and SNe IIP.

The best way to calibrate the thermalisation parame-ter in STELLA is to compare to simulations calculated withadvanced radiative transfer codes that do not use the sim-plified treatment of line opacity. These codes may allow pho-tons to behave consistently with a set of calculated Non-LTE

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Treatment of line opacity and SN light curves 3

level populations, or at least without the requirement of afree parameter governing absorption and scattering. For SNeType Ia we chose the widely-used W7 model to carry outcomparison to the existing simulations done with the ARTIScode Kromer & Sim (2009). Compared to STELLA, ARTIS hasthe following advantages:

(i) Each line is treated individually, without opacity bin-ning.

(ii) There is no concept of the thermalisation parameter.Radiation–matter interactions are always treated in detail(i.e. statistical equilibrium), e.g. the full macro-atom ma-chinery is used to model fluorescence.

(iii) Although it does not use Non-LTE level populations,ARTIS calculates a Non-LTE ionisation balance. It does thisby recording detailed photoionisation rate estimators for theground level of all ions, and approximating photoionisationrates of excited states by assuming that they scale with theground level rates by the same factor as in LTE. Within anionisation stage, the level populations are calulated fromthe Boltzmann distribution at the radiation temperature.

On top of that, we test our calibration based on com-parison to observations, since ARTIS qualitatively repro-duces realistic behaviour of the spectral energy distribu-tion, i.e. broad band fluxes. For SNe IIP and SNe-IIpec,there are no well-accepted models. Therefore, we calibratethe STELLA thermalisation parameter via comparison to ob-servations, assuming that sophisticated codes like CMFGEN,ARTIS, PHOENIX and others reproduce colours for observednormal SNe IIP and SN 1987A (i.e. SNe-IIpec) well.

3 DISCUSSION

3.1 Application to SN Ia

We analyse the behaviour of the numerically computedSTELLA light curves for comparison to simulations done withthe radiative transfer code ARTIS, and observations of thenormal SN Ia 2005cf (Pastorello et al. 2007). These lightcurves are shown in Figure 1.

The W7 model gives a reasonable match to the generalcolour evolution of SN2005cf, although there are significantdiscrepancies. Even though the W7 model is one choice ofmany candidate explosion models, and would therefore notbe expected to match all of the features of SN2005cf, we stilluse this combination of the theoretical model and observeddata for our study. Note that we mapped into STELLA the hy-drodynamical and chemical profiles of the model W7 whichis exactly the same model used for ARTIS by Kromer & Sim(2009). Neither ARTIS nor STELLA simulations can ex-plain the detailed behaviour of SN2005cf, particularly, thesecond R and I maxima (see Figure 1). There are a numberof reasons for this. The ARTIS curves clearly demonstratethe appearance of the second maximum which occurs a bitearlier than the observed maximum in SN2005cf, and whichis brighter (I ) than the observed one. It has been shown thatnon-LTE (either approximate or accurate) radiative trans-fer may better reproduce the second maxima (Blondin et al.2011, 2015). Again, we emphasise that the model W7 it-self is one particular choice of explosion model, and even

theoretically-perfect radiative transfer for this model willnot necessarily match observed SNe Ia spectra. However, anumber of explosion models can not reproduce the secondmaximum at all (Ohlmann et al. 2014). It could be that aslightly lower-mass model which host different thermody-namical conditions for the iron-group elements may bet-ter reproduce the maxima occurence (Blondin et al. 2017,2018). The yields of individual species like Sc, Ti, Cr, Fe,and radioactive 56Ni, and their distribution in the ejecta alsostrongly affect the location of the maxima (Blondin et al.2013; Shen et al. 2018). As for STELLA, the code treats alimited number of species: H, He, C, N, O, Ne, Na, Mg, Al,Si, S, Ar, Ca, stable Fe, stable Ni, stable Co, radioactive56Ni, and considers 150,000 lines in the standard settings(Kurucz & Bell 1995). Hence line lists for individual specieslike Sc, Ti, Cr are not included while they strongly con-tribute to line opacity. Hence, STELLA poorly reproducesthe second maxima in general, although qualitatively repre-sents colour evolution comparable with more sophisticatedcodes even with the limited number of lines (Woosley et al.2007). Certainly, new physics and extentions to the line listin STELLA will be important areas of progress in futureversions of the code.

In Figure 1, we present the results of our simulations ofthe model W7 done with STELLA and a range of values forthe thermalisation parameter ε. This allows us to explorethe influence of different contributions to absorption andscattering in bound-bound transitions. We also superposethe results of the simulation by Kromer & Sim (2009) andthe observed normal SN Ia 2005cf (Pastorello et al. 2007).We show six curves computed with STELLA with six corre-sponding thermalisation parameters: 100% absorption (thelabel “Pure Absorption”), 90% absorption + 10% scattering(“90% Abs 10% Scat”), 80% absorption + 20% scattering(“80% Abs 20% Scat”), 50% absorption + 50% scattering(“50% Abs 50% Scat”), 10% absorption + 90% scattering(“10% Abs 90%Scat”), and 100% scattering (“Pure Scat-tering”). Hence, we explore the variation of thermalisationparameter between 0 (pure scattering) and 1 (pure absorp-tion). The general property of the STELLA light curves withdifferent values for ε is overestimated U magnitude, how-ever presumably this is an intrinsic property of the givenmodel. While U magnitude is almost independent on thechoice of thermalisation parameter with moderate contribu-tion of scattering (ε = 0.8− 1), it has a stronger impact forR and I magnitudes. ε = 0.5 provides too broad light curvein B, therefore, ε = 0.8−1 is more realistic. Light curves cal-culated with larger contribution of scattering (ε < 0.5) arefully incompatible with the real observed data for SNe Ia. Itis worth noting that light curves with ε = 0.1 have two pro-nounced maxima which are proven observational propertyof SNe Ia. We discuss this aspect below.

There are two major contributors to opacity in the W7model – iron and calcium (considered by STELLA). As dis-cussed in Kasen (2006), considering calcium lines as purelyabsorptive results in overestimated I magnitude. Therefore,SEDONA treats calcium lines as purely scattering, whileiron lines tend to be purely absorptive. STELLA applies ther-malisation parameter equally to all elements with no excep-tions. With the currently implemented line list, STELLA doesnot resolve two maxima in I band. Surprisingly, the casewith 90% scattering does exhibit two maxima in the STELLA

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Figure 1. The broad-band magnitudes for the W7 model for different cases of the ratio between absorption and scattering computedwith STELLA, broad-band magnitudes for the model W7 computed with ARTIS (Kromer & Sim 2009), and the observed magnitudesof SN 2005cf (Pastorello et al. 2007). “Abs” stands for absorption fraction, and “Scat” stands for scattering fraction. “Pure Absorption”means 100% of absorption, and “Pure Scattering” means 100% of scattering.

light curve, however both maxima are underestimated in lu-minosity. This means that at least some strong lines, i.e.most likely calcium lines, have to be treated with sufficientlylow thermalisation parameter, e.g. ε = 0.1, i.e. almost purelyscattering. As a consequence, U band light curve for thecase of ε = 0.1 is much closer to the observed magnitude ofSN2005cf. Nevertheless, the widths of the light curves in allbands are too large and inconsistent with the observed data.

In Figure 2, we show STELLA SEDs for the model W7 fordifferent values of thermalisation parameter at day 15 andday 40. SED maxima for the cases with 100%, 90%, 80%,and 50% absorption are very close to each other at day 15,although the exact maximum wavelength differs by 300 A.At a later epoch, day 40, the cases of 100%, 90%, and 80%are almost identical, while the case of 50% provides rela-tively bluer spectrum. Therefore, we rule out values for ther-malisation parameter below 0.8. If comparing our SEDs tothe result of spectral synthesis simulations by SEDONA (Fig. 5,Kasen et al. 2006) and ARTIS (Fig. 6, Kromer & Sim 2009),we conclude that spectral maximum lies around 3500 A andaround 6000 – 6500 A at day 15 and day 40, respectively, i.e.thermalisation parameter tends to be close to unity.

We carry out a quantative analysis, particularly, we cal-culated the linear Pearson’s correlation coefficient with the95% confidence interval to pick the most plausible value for

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Figure 2. Spectral energy distribution for the model W7 withdifferent values of thermalisation parameter: 1 (blue dashed), 0.9(red), 0.8 (yellow dashed), 0.5 (green), 0.1 (magenta dotted), and0 (black dash-dotted), at day 15 and day 40. Labels have the samemeaning as in Figure 1.

the thermalisation parameter (e.g., for the correlation be-tween 2005cf and the STELLA curves). We have confirmedthat our findings with the correlation method are in agree-

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Treatment of line opacity and SN light curves 5

ment with several alternative statistical methods to quan-tity the best-fit (for details, see Appendix A). The observeddata and the synthetic light curves are sampled on differentgrids of time points. Therefore, we first applied a 10-th or-der polynomial regression or smoothing spline (using Matlabstandard libraries) with the least standard deviation crite-rion. We then evaluated the resulting polynomial (or spline)on 100 equidistant time points. We therefore converted eachlight curve into a vector of a uniform length, discretised ontoa regular time grid. The χ–squared test evaluates statisticaldependence between standard deviations under the condi-tion of normally distributed residuals. χ–squared test wasnot considered because (1) given arrays are very limited intime, (2) the data points within a given array are not inde-pendent, and (3) residuals do not have normal distribution.Therefore, we choose the estimate of the correlation coeffi-cient in the linear regression approach as a measure of sta-tistical coherence between vectors. This method was appliedto every curve in the study. Correlation analysis, i.e. evalua-tion of statistical dependence, was based on the calculationof the Pearson’s correlation coefficient in the frame of thelinear regression model (Afifi & Azen 1979; Aivazyan et al.1985):

r =

N∑

i=1

(xi − 〈x〉)(yi − 〈y〉)√σxσy

, (1)

where 〈x〉 and 〈y〉 are the means, and σx and σy are thestandard deviation for light curve vectors x and y, respec-tively. The correlation analysis was carried out for each pairof curves: U-band 2005cf and U-band STELLA curve, B-band 2005cf and B-band STELLA curve and so on. Theinterval is chosen between day 2 and day 85 for STELLA-2005cf correlation. The same method was applied for pairsof ARTIS and STELLA curves. The interval is 2-72 daysfor ARTIS-STELLA correlation. In each pair of curves, theinterval is chosen to be the longest time interval over whichboth curves are defined.

We analyse correlation between STELLA and ARTIS,and STELLA and SN2005cf. The closest correlation betweenSTELLA and ARTIS is observed for the cases of 100%, 90%,and 80% of absorption with the coefficients 0.975, 0.953,0.980, 0.899–0.919, and 0.899–0.918 for U, B, V, R, and I,correspondingly. Since we look for the choice which includesat least some scattering, we conclude that the best casesare 80% and 90% absorption. We apply the same procedureto correlate STELLA and SN2005cf. The resulting correla-tion coefficients show the tight correlation for the cases ofthermalisation parameter ε = 0.5 for U band with coeffi-cient 0.987, and for ε = 0.8− 1 with coefficient 0.995, 0.999,0.894–0.909, and 0.862-0.878 for B, V, R, and I, respectively.To conclude, we suggest to set thermalisation parameter be-tween 0.8 and 0.9, and more precisely ε = 0.9 for consistencywith SNe II which we discuss in the sections below.

We conclude that correlation coefficients derived forSTELLA–SN 2005cf are the same as for STELLA–ARTIS. There-fore, we follow the same procedure for SNe type IIP and 87A-like, namely we analyse the correlation between the STELLA

light curves and observed data to define the plausible ther-malisation parameter. On top of that, calibrating the codeparameter on the real observations complements our anal-

ysis, and makes future numerical simulations with STELLA

even more robust.

3.2 Application to SN 1987A

Modelling of the progenitor of SN 1987A is still an openquestion, and there is no model that accurately repro-duces all observed features. There are single star mod-els (Woosley 1988; Shigeyama & Nomoto 1990), and bi-nary models (Menon & Heger 2017; Urushibata et al. 2018;Ono et al. 2020). Among the difficuties is the problem ofproducing a blue supergiant model at LMC metallicity. Sin-gle star models are computed with artificially reduced metalcontent to get a relatively compact (∼ 50 R⊙) progeni-tor. Models with the low metal content, i.e. low metallic-ity, provide too blue colours, particularly, U band magni-tude is overestimated (Blinnikov 1999). At the same time,binary merger models by Menon & Heger (2017) have suffi-cient metals because of accretion from the companion, andthey are more promising as an explanation for, particularly,the U magnitude of SN 1987A. Therefore, we pick up oneof their recent binary models, 16-7b (Menon et al. 2019), tocalibrate the thermalisation parameter in STELLA.

The light curves taken with different values of thermal-isation parameter are shown in Figure 3. Figure 3 showsbolometric light curves (upper left) which are in acceptableagreement with observations for the choosen exposion en-ergy of 2.33 foe. U and B magnitudes are the most affectedby variation in the thermalisation parameter, while variationin V,R, and I magnitudes is not large for different parametervalues. We show U, B and I magnitudes for demonstration.We note that we do not pay attention to the radioactivetail, but mostly concentrate on the photospheric phase, i.e.before approximately day 120. STELLA does not providereliable broad-band magnitudes after this epoch, since theSN ejecta becomes semi-transparent, and Non-LTE treat-ment is requred. For quantative analysis, we apply the samecorrelation procedure as discussed in Section 3.1. Hence wecalculate the correlation coefficients between the numericalSTELLA light curves in broad bands and broad band magni-tudes of SN1987A (Menzies et al. 1987). Light curves com-puted with ε = 0.8, 0.9 and 1 evolve close to each other.However, taking into account the necessity of a contribu-tion from resonant scattering, we conclude that ε has to belower than 1, e.g. 0.8 – 0.9. From the correlation analysis, themost suitable value for the thermalisation parameter lies be-tween 0.8 and 1, if we ignore the a priori irrelevant valuescorresponding to pure scatterring and the case with 10%of absorption. The correlation coefficients are: 0.911–0.926,0.872–0.899, 0.9–0.917, 0.936, and 0.947, for U, B, V, R, andI, respectively.

Additionally, we did a test using the spectral synthe-sis code ARTIS for the model B15-2 (Utrobin et al. 2015)which is similar to the model 16-7b in our study except fora zero metallicity. We run the model with a low numberof photon packets to explore the contribution by resonancescattering and true absorption. The photon packets experi-ence 252130, 265155, 162141, and 3152 line interactions intimesteps at day 60, day 80, day 100, and day 140, while4943, 5324, 3984, and 159 of those are pure scattering (i.e.no wavelength change). Hence, the thermalisation parame-ter, i.e. the ratio between the number of inelastic interactions

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Figure 3. Bolometric light curves and U, B and I broad-band magnitudes for the model 16-7b for different cases of the ratio betweenabsorption and scattering. Labels have the same meaning as in Figure 1.

(absorption) and total number of interactions is: 0.98, 0.98,0.975, and 0.95 for corresponding epochs. These numbers arevery close to unity, i.e. they show the dominant contributionof absorption to the line opacity. However, this also demon-strates the inevitability of a non-zero scattering fraction inthe line opacity.

From our analysis, we conclude that the thermalisationparameter for hydrogen-rich SNe like SN1987A falls into thesame interval as for Ni-powered SNe Ia, i.e. 0.8 – 0.9, withthe highest plausible value of 0.9.

3.3 Application to SNe IIP

The model L15 is the accepted model which reproducesbolometric properties of SN1999em (Elmhamdi et al. 2003;Utrobin 2007; Utrobin et al. 2017). In Figure 4, we showbolometric light curves and broad-band magnitudes for themodel L15 for different cases of the ratio between absorptionand scattering. Firstly, we present bolometric light curves forthe subset of the model L15 (the upper left plot in Figure 4)to demonstrate that the model is indeed sufficient to explainthe bolometric luminosity of SN1999em. As in previous sec-tions, we vary the thermalisation parameter between 0 and1 for the model L15. The overall differences between bolo-metric light curves are not large. There is a tiny variation inearly plateau luminosity (maximum 0.05 dex), while thereis a noticable difference in the behaviour of the transition

of the light curve to the radioactive tail. Nevertheless, thetreatment of lines is indeed not important for the bolometricproperties at earlier epoch because lines contribute less sig-nificantly than the continuum opacity during the early phaseand especially during the relaxation after shock breakout.However, the lines start playing a significant role at day 20for our model L15 (see Figure 4). In any case, the bolomet-ric light curve is rather insensitive to different degrees ofthermalisation in lines because it reflects the overall energybudget (which is conserved), while different thermalisationvalues only cause redistribution of energy between parts ofthe spectrum.

With the pure scattering line opacity, U and B remainblue for the entire plateau. Blue photons from the decay of56Ni are accumulated in the region where they are born. Lineopacity is much higher for blue photons than for redder pho-tons, with the effect that blue photons require a longer timeto diffuse through the optically-thick medium. Therefore,scattering-dominated line opacity provides a larger fractionof blue photons remaining in the inner region of the SNejecta. Assuming SN 1999em to be a normal SN IIP, U

fades during the plateau phase with the slope 4–5 magsduring 100 days, and B magnitude drops with the slope2 mags during this period. This is consistent with our mod-elled curves with scattering fraction of 0–20 %. Observationsby Faran et al. (2014), Valenti et al. (2016) and Szalai et al.(2019) also confirm these slopes and present a large set of

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Figure 4. Bolometric light curves and broad-band magnitudes for the model L15 for different cases of the ratio between absorption andscattering. We superpose observed SN 1999em as crosses (Elmhamdi et al. 2003). Labels have the same meaning as in Figure 1 exceptthe magenta curve, which stands now for the case of 20 % of absorption and 80 % of scattering.

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2000 4000 6000 8000 10000 12000 [Å]

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Figure 5. Spectral energy distribution for the model L15 forcases of 0 % (blue dashed), 20 % (red), 80 % (green dashed), and100 % (magenta) of scattering at day 50 and day 130. Labels havethe same meaning as in Figure 4.

SNe IIP which show standard decline of about 2 mags in B

band during 100 days. R and I light curves are not stronglyaffected by the value of the thermalisation parameter. Weconclude that introducing a larger fraction of scattering to

0 20 40 60 80 100 120 140 160Time [days]

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00 K

]

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Figure 6. Colour temperature evolution for the model L15 forcases of 0 % (blue dashed), 20 % (red), 80 % (green dashed), and100 % (magenta) of scattering. Labels have the same meaning asin Figure 4.

the line opacity leads to long-lasting blue colours which isnot supported by observations of normal SNe IIP.

We follow the same correlation procedure as we applied

MNRAS 000, 1–9 (2020)

8 A. Kozyreva et al.

for SNe Ia and SNe 1987A-like. We found that the lightcurves with ε = 0.8, 0.9, and 1 are the most correlated withthe observed broad-band magnitudes of SN 1999em. Thecorrelation coefficients are: 0.98, 0.992, and 0.992 for U, B,and V, respectively.

In Figure 5, we show spectral energy distributions forthe model L15 for cases of 0 %, 20 %, 80 %, and 100 %of scattering at day 50 and day 130. There is an extrablue flux for those cases where scattering fraction is higher,which leads to higher colour temperature. Figure 6 showscorresponding colour temperature evolution for these cases.While pure absorption (what means 0 % scattering) and20 % scattering cases are suitable for normal SNe IIP (seee.g., Bersten & Hamuy 2009), curves with larger scatter-ing contribution are totally unrealistic, because of relativelytoo-blue colour at the end of their plateau.

To conclude, we find the value ε = 0.9 provides the bestmatch to colours in normal SN IIP.

4 CONCLUSIONS

We explored the treatment of lines in the hydrodynamicsradiative transfer code STELLA. Previously, all lines in thecode were considered as purely absorptive, i.e. a photonwas immediately thermalised on interaction with matter. Weanalysed the impact of introduction scattering into the linetreatment using three reference models, W7 (Nomoto et al.1984), 16-7b (Menon & Heger 2017; Menon et al. 2019), andL15 (Limongi et al. 2000; Utrobin et al. 2017), to illustratethe impact of different degree of thermalisation in lines onthe behaviour of light curves and SEDs of normal SNe Ia,SNe IIpec, and normal SNe IIP. We analysed light curves inthe broad bands in the context of the more sophisticatedsimulations done with the code ARTIS, and well-observedSN2005cf, SN1987A, and SN1999em.

We found that the most suitable value for the thermal-isation parameter ε, i.e. the relative contribution of absorp-tion to overall line opacity, lies between 0.8 and 0.9 for thethree types of SNe considered. The scattering due to linesshould be less than 10–20% to prevent blue flux from ex-ceeding observed SNe.

Our recommendation is to use ε = 0.9 in all futuresimulations that will be done with the code STELLA which isa part of the latest MESA release (Paxton et al. 2018).

ACKNOWLEDGMENTS

We thank the referee Prof. Saurabh W. Jha for his interestin our study and careful analysis of our methodology andresults that improved the paper. AK is supported by theAlexander von Humboldt Foundation. LS acknowledges sup-port from STFC through grant, ST/P000312/1. SB and PBare sponsored by grant RSF18-12-00522. Some of this workwas performed using the Cambridge Service for Data DrivenDiscovery (CSD3), part of which is operated by the Uni-versity of Cambridge Research Computing on behalf of theSTFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRACcomponent of CSD3 was funded by BEIS capital fundingvia STFC capital grants ST/P002307/1 and ST/R002452/1and STFC operations grant ST/R00689X/1. DiRAC is part

of the National e-Infrastructure. The authors thank Vik-tor Utrobin, Thomas Janka for discussions which initiatedthis study, and Markus Kromer and Stuart Sim for clarifi-cations about ARTIS simulations, Anders Jerkstrand, IldarKhabibullin and Marat Potashov for overall discussions.

DATA AVAILABILITY

The data computed and analysed forthe current study are available via linkhttps://wwwmpa.mpa-garching.mpg.de/ccsnarchive/data/Kozyreva2018/index.html.

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APPENDIX A: ADDITIONAL STATISTICS

We carried out additional statistical tests to verify the re-sults of the correlation method that we used in Section 3.

Let us consider the pairs of arrays from our study,namely, the B -band magnitudes calculated with ARTIS andSTELLA (with different degree of thermalisation). In Fig-ure A1, we demonstrate the pairs of arrays: ARTIS B -bandmagnitude versus STELLA B -band magnitude computed withdifferent value of the thermalisation parameter. We calcu-lated the linear regression coefficient for each pair, assum-ing Y = A×X +B, where X is ARTIS magnitude and Y isSTELLA magnitude.

We consider the statistic:

s =∑

i

[(yi − 〈y〉)− (xi − 〈x〉)] 2 (A1)

and slightly modified statistic:

sm =∑

i

[

yi − 〈y〉〈y〉 − xi − 〈x〉

〈x〉

] 2

. (A2)

The minimal value for these statistics will indicate the bestmatch between arrays X and Y .

In Table A1, we list the resulting coefficients. Here, Ais linear regression coefficient, r is linear correlation coeffi-cient (see Equation 1 in Section 3.1), and two coefficientss and sm counting for the proposed additional statistics.The maximal correlation coefficient correspond to minimals and sm coefficients. We note though that maximal r andminimal s/sm do not necessarily correspond to the maximallinear regression coefficient (its closure to 1). We highlight inbold face the best match according to all considered statis-tics. Hence, we rely on the maximal correlation coefficientin the main body of the paper as the diagnostic for the bestmatch between light curves and the selection of the best

value for the thermalisation parameter. We list the coeffi-cients for the combination ARTIS–STELLA in the U, V, R,and I broad bands in Table A2.

In Figures A2, A3, and A4, we demonstrate the B -bandSTELLA magnitudes and corresponding magnitudes of theSN2005cf, SN1987A, and SN1999em. Tables A3, A4, andA5 contain calculated coefficients for the cases considered inthe paper: the model W7 and SN2005cf, the model 16-7band SN1987A, and the model L15 and SN1999em.

This paper has been typeset from a TEX/LATEX file prepared bythe author.

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Data: r=0.86451; s=30.1625; s_m=0.11139Linear Fit: Y=0.88609*X-0.6928395% Prediction Interval

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Data: r=0.82496; s=38.2734; s_m=0.12592Linear Fit: Y=0.52891*X-8.112495% Prediction Interval

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Data: r=0.94696; s=11.1918 s_m=0.036937Linear Fit: Y=0.89142*X-1.718195% Prediction Interval

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Data: r=0.9535; s=10.0714; s_m=0.033763Linear Fit: Y=0.95307*X-0.5812395% Prediction Interval

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Data: r=0.9537; s=10.2234; s_m=0.034492Linear Fit: Y=0.96902*X-0.2873195% Prediction Interval

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Data: r=0.95266; s=10.5997 s_m=0.035918Linear Fit: Y=0.97752*X-0.124795% Prediction Interval

Figure A1. The B-band magnitudes of the W7 model computed with STELLA using different assumed ratios between absorption andscattering, and the W7 model computed with ARTIS (Kromer & Sim 2009).

Table A1. The linear regression coefficient A, linear correlation coefficient r, and the statistics s and sm for the pairs of light curvesin B broad band computed with STELLA and ARTIS. The line in bold face corresponds to the maximum correlation coefficient andminimum s and sm coefficients.

A r s sm

BARTIS BSTELLA000A 0.88609 0.86451 30.1625 0.11139BARTIS BSTELLA010A 0.52891 0.82496 38.2734 0.12592BARTIS BSTELLA050A 0.89142 0.94696 11.1918 0.036937BARTIS BSTELLA080A 0.95307 0.95350 10.0714 0.033763BARTIS BSTELLA090A 0.96902 0.95370 10.2234 0.034492

BARTIS BSTELLA100A 0.97752 0.95266 10.5997 0.035918

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Data: r=0.96535; s=11.2696; s_m=0.040054Linear Fit: Y=0.90813*X-0.4275495% Prediction Interval

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Data: r=0.88695; s=59.0431; s_m=0.20973Linear Fit: Y=0.44703*X-9.624495% Prediction Interval

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Data: r=0.99166; s=8.7488; s_m=0.031619Linear Fit: Y=0.79334*X-3.597995% Prediction Interval

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Data: r=0.99517; s=4.5631 s_m=0.016045Linear Fit: Y=0.85616*X-2.447295% Prediction Interval

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Data: r=0.99529; s=3.872; s_m=0.013464Linear Fit: Y=0.87199*X-2.157395% Prediction Interval

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Data: r=0.99515; s=3.4854; s_m=0.011987Linear Fit: Y=0.88324*X-1.946395% Prediction Interval

Figure A2. The B-band magnitudes of the W7 model computed with STELLA using different assumed ratios between absorption andscattering, and observations of SN 2005cf.

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Treatment of line opacity and SN light curves 11

Table A2. The same as in Table A1 for the W7 U,V,R,I-band STELLA magnitudes and ARTIS magnitudes.

A r s sm

UARTIS USTELLA000A 0.64213 0.97445 40.67760 0.13968UARTIS USTELLA010A 0.44048 0.80076 114.35609 0.41186UARTIS USTELLA050A 0.81061 0.95934 25.15613 0.09365UARTIS USTELLA080A 0.88321 0.97406 15.11910 0.05550UARTIS USTELLA090A 0.90048 0.97608 13.59926 0.04947UARTIS USTELLA100A 0.91200 0.97707 12.81158 0.04621

VARTIS VSTELLA000A 0.88609 0.86451 30.1625 0.11139VARTIS VSTELLA010A 0.52891 0.82496 38.2734 0.12592VARTIS VSTELLA050A 0.89142 0.94696 11.1918 0.036937VARTIS VSTELLA080A 0.95307 0.95350 10.0714 0.033763VARTIS VSTELLA090A 0.96902 0.95370 10.2234 0.034492VARTIS VSTELLA100A 0.97752 0.95266 10.5997 0.035918

RARTIS RSTELLA100A 0.84774 0.86779 27.72292 0.09397RARTIS RSTELLA100A 0.25819 0.51068 79.17357 0.24369RARTIS RSTELLA100A 0.55486 0.84413 34.52384 0.10564RARTIS RSTELLA100A 0.61134 0.89904 25.67382 0.07864RARTIS RSTELLA100A 0.62611 0.91026 23.65941 0.07251RARTIS RSTELLA100A 0.63532 0.91865 22.23886 0.06819

IARTIS I STELLA100A 0.84681 0.78094 39.04812 0.13263IARTIS I STELLA100A 0.26291 0.47073 63.65931 0.19110IARTIS I STELLA100A 0.63093 0.84502 23.93898 0.07053IARTIS I STELLA100A 0.69274 0.89938 16.82663 0.04937IARTIS I STELLA100A 0.70765 0.91048 15.28726 0.04483IARTIS I STELLA100A 0.71676 0.91804 14.25680 0.04180

Table A3. The same as Table A1 for the W7 STELLA broad-band magnitudes and SN 2005cf magnitudes.

A r s sm

U sn2005cf U STELLA000A 0.81004 0.96627 17.38413 0.06189U sn2005cf U STELLA010A 0.67581 0.93254 36.48919 0.14668U sn2005cf U STELLA050A 1.05999 0.99128 4.92997 0.01490U sn2005cf U STELLA080A 1.13471 0.99130 8.58731 0.02674U sn2005cf U STELLA090A 1.15321 0.99087 10.11125 0.03232U sn2005cf U STELLA100A 1.16688 0.99045 11.40321 0.03729

B sn2005cf B STELLA000A 0.90813 0.96535 11.26962 0.04005B sn2005cf B STELLA010A 0.44703 0.88695 59.04310 0.20973B sn2005cf B STELLA050A 0.79334 0.99166 8.74879 0.03162B sn2005cf B STELLA080A 0.85616 0.99517 4.56309 0.01604B sn2005cf B STELLA090A 0.87199 0.99529 3.87200 0.01346B sn2005cf B STELLA100A 0.88324 0.99515 3.48541 0.01199

V sn2005cf V STELLA000A 1.22312 0.93853 20.54001 0.09563V sn2005cf V STELLA010A 0.30969 0.82297 42.53423 0.13506V sn2005cf V STELLA050A 0.65986 0.99167 10.02148 0.03177V sn2005cf V STELLA080A 0.72819 0.99885 6.11644 0.01934V sn2005cf V STELLA090A 0.74629 0.99915 5.31970 0.01679V sn2005cf V STELLA100A 0.76039 0.99922 4.75023 0.01496

R sn2005cf R STELLA000A 1.24223 0.91316 25.27996 0.10852R sn2005cf R STELLA010A 0.24682 0.55332 48.70329 0.15306R sn2005cf R STELLA050A 0.60890 0.88714 17.48942 0.05509R sn2005cf R STELLA080A 0.67632 0.92815 12.31342 0.03897R sn2005cf R STELLA090A 0.69377 0.93631 11.15055 0.03535R sn2005cf R STELLA100A 0.70649 0.94276 10.26031 0.03257

I sn2005cf I STELLA000A 1.45465 0.90383 32.08675 0.12960I sn2005cf I STELLA010A 0.09844 0.28903 43.30838 0.13566I sn2005cf I STELLA050A 0.54445 0.82089 16.53865 0.05180I sn2005cf I STELLA080A 0.61584 0.86225 13.11988 0.04129I sn2005cf I STELLA090A 0.63418 0.87236 12.25684 0.03864I sn2005cf I STELLA100A 0.64790 0.87824 11.70627 0.03694

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Data: r=0.88465; s=56.3978; s_m=0.31547Linear Fit: Y=0.65784*X-5.059995% Prediction Interval

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Data: r=0.33557; s=217.8346; s_m=1.1748Linear Fit: Y=0.16986*X-11.927495% Prediction Interval

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Data: r=0.77894; s=97.7511 s_m=0.53489Linear Fit: Y=0.50217*X-6.948995% Prediction Interval

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Data: r=0.87192; s=60.2062; s_m=0.32705Linear Fit: Y=0.65872*X-4.607895% Prediction Interval

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Linear Fit of Data with 95% Prediction Interval

Data: r=0.88908; s=53.2302; s_m=0.28808Linear Fit: Y=0.68371*X-4.2195% Prediction Interval

-15 -14.5 -14 -13.5 -13 -12.5 -12sn87a_B

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Linear Fit of Data with 95% Prediction Interval

Data: r=0.89881; s=48.0397 s_m=0.25931Linear Fit: Y=0.71808*X-3.692595% Prediction Interval

Figure A3. The B-band magnitudes for the 16-7b model computed with STELLA using different ratios between absorption andscattering, and observations of SN 1987A.

Table A4. The same as Table A1 for the 16-7b STELLA broad-band magnitudes and SN 1987A magnitudes.

A r s sm

U87A USTELLA000A 0.06837 0.13050 317.80879 2.28892U 87A USTELLA010A 0.08695 0.13709 343.13217 2.44217U 87A USTELLA050A 0.76728 0.85858 73.76258 0.56205U 87A USTELLA080A 1.03185 0.91171 60.67765 0.43050U 87A USTELLA090A 1.08594 0.92123 60.80450 0.43279U 87A USTELLA100A 1.13518 0.92595 64.97415 0.46718

B 87A BSTELLA000A 0.65784 0.88465 56.3978 0.31457B 87A BSTELLA010A 0.16986 0.33557 217.8346 1.1748B 87A BSTELLA050A 0.50217 0.77894 97.7511 0.53489B 87A BSTELLA080A 0.65872 0.87192 60.2062 0.32705B 87A BSTELLA090A 0.68371 0.88908 53.2302 0.28808B 87A BSTELLA100A 0.71808 0.89881 48.0397 0.25931

V 87A VSTELLA000A 0.79803 0.92585 35.4874 0.15782V 87A VSTELLA010A 0.45557 0.74768 111.1707 0.50002V 87A VSTELLA050A 0.58864 0.85189 72.5256 0.32627V 87A VSTELLA080A 0.67421 0.90059 51.2285 0.22959V 87A VSTELLA090A 0.68007 0.91032 47.8315 0.21402V 87A VSTELLA100A 0.69792 0.91697 44.3254 0.19796

R 87A RSTELLA000A 0.91404 0.92960 23.77577 0.09832R 87A RSTELLA010A 0.82889 0.92540 24.76832 0.10066R 87A RSTELLA050A 0.81238 0.92252 25.83081 0.10519R 87A RSTELLA080A 0.84719 0.93368 22.09643 0.08996R 87A RSTELLA090A 0.84185 0.93614 21.42282 0.08721R 87A RSTELLA100A 0.84607 0.93794 20.83242 0.08481

I 87A I STELLA000A 0.99412 0.94056 20.47483 0.08229I 87A I STELLA010A 0.96658 0.96296 11.81486 0.04632I 87A I STELLA050A 0.88718 0.94229 17.80415 0.06965I 87A I STELLA080A 0.90315 0.94723 16.32774 0.06381I 87A I STELLA090A 0.90073 0.94690 16.42541 0.06422I 87A I STELLA100A 0.89997 0.94716 16.34396 0.06393

MNRAS 000, 1–9 (2020)

Treatment of line opacity and SN light curves 13

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Linear Fit of Data with 95% Prediction Interval

Data: r=0.94571; s=19.5134; s_m=0.11392Linear Fit: Y=0.84265*X-3.397595% Prediction Interval

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Linear Fit of Data with 95% Prediction Interval

Data: r=0.98787; s=12.9306; s_m=0.092531Linear Fit: Y=0.75975*X-4.386195% Prediction Interval

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Linear Fit of Data with 95% Prediction Interval

Data: r=0.99081; s=4.1845; s_m=0.030831Linear Fit: Y=0.91161*X-2.024995% Prediction Interval

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Linear Fit of Data with 95% Prediction Interval

Data: r=0.99372; s=31.8407 s_m=0.16136Linear Fit: Y=1.3908*X5.35395% Prediction Interval

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Linear Fit of Data with 95% Prediction Interval

Data: r=0.99398; s=38.751; s_m=0.20392Linear Fit: Y=1.4366*X6.070895% Prediction Interval

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Linear Fit of Data with 95% Prediction Interval

Data: r=0.99425; s=46.5808; s_m=0.25409Linear Fit: Y=1.4835*X6.808595% Prediction Interval

Figure A4. The B-band magnitudes for the L15 model computed with STELLA using different ratios between absorption and scattering,and observations of SN 1999em.

Table A5. The same as Table A1 for the L15 STELLA broad-band magnitudes and SN 1999em magnitudes.

A r s sm

U sn99em USTELLAL15 000A 0.30067 0.97527 80.26059 0.44725U sn99em USTELLAL15 010A 0.67449 0.98157 20.02893 0.13150U sn99em USTELLAL15 020A 0.76093 0.97187 14.82002 0.09496U sn99em USTELLAL15 080A 0.93246 0.98199 5.97510 0.03312U sn99em USTELLAL15 090A 0.95300 0.98158 5.95111 0.03151U sn99em USTELLAL15 100A 0.97224 0.98208 5.78599 0.02953

B sn99em BSTELLAL15 000A 0.84265 0.94571 19.51341 0.11392B sn99em BSTELLAL15 010A 0.75975 0.98787 12.93062 0.09253B sn99em BSTELLAL15 020A 0.91161 0.99081 4.18450 0.03083B sn99em BSTELLAL15 080A 1.39078 0.99372 31.84070 0.16136B sn99em BSTELLAL15 090A 1.43664 0.99398 38.75098 0.20392B sn99em BSTELLAL15 100A 1.48351 0.99425 46.58084 0.25409

V sn99em VSTELLAL15 000A 1.13095 0.98045 7.21903 0.02631V sn99em VSTELLAL15 010A 0.86758 0.98447 4.36066 0.02386V sn99em VSTELLAL15 020A 0.95116 0.98997 2.18841 0.01139V sn99em VSTELLAL15 080A 1.26760 0.99398 9.58007 0.03485V sn99em VSTELLAL15 090A 1.30034 0.99346 11.83005 0.04448V sn99em VSTELLAL15 100A 1.33826 0.99367 14.437937 0.05595

MNRAS 000, 1–9 (2020)